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16.346 Astrodynamics
Fall 2008
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Lecture 23 Estimation of Position and Velocity in Space Navigation
Recall the Definitions
Deviation in quantity measured:
h
Measurement vector: b =
0
State vector deviation:
δq
δr(t)
δx(t) =
δv(t)
δq = bT δx
Fundamental relationship:
Φ(tn , tn−1 ) = Φn,n−1
State transition matrix:
State vector deviations at tn and tn−1 :
Fundamental relationship:
δxn and δxn−1
δxn = Φn,n−1 δxn−1
Effect at tn of observation made at tn−1
δq(tn−1 ) = δqn−1 = bnT−1 δxn−1 = bnT−1 Φ−1
n,n−1 δxn
Recursive Formulation of the Navigation Algorithm
∗n = δx
n + w(δq − δq )
δx
n
where δq = bT δx
n = Φn,n−1 δx
n−1
and δx
Propagating the Covariance Matrix P and the Error Transition Matrix W
• Using the state transition matrix
n−1 = δxn−1 + en−1
δx
n = Φn,n−1 δx
n−1
δx
en = Φn,n−1 en−1
=⇒
T
T
eT
n = en−1 Φn,n−1
T
T
en e T
n = Φn,n−1 en−1 en−1 Φn,n−1
δxn = Φn,n−1 δxn−1
Hence
T
Pn = Φn,n−1 Pn−1 Φn,n−1
and Wn = Φn,n−1 Wn−1
• Using differential equations
dx
= Fx
dt
=⇒
de
= Fe and
dt
de T
= eT FT
dt
Hence
dP
= FP + PF T
dt
16.346 Astrodynamics
and
Lecture 23
dW
= FW
dt
Encke’s Method of Orbital Integration
#9.4
Deviations from the Osculating Orbit
Define
Then since
r(t0 ) = rosc (t0 )
r(t) = rosc (t) + δ(t)
v(t0 ) = vosc (t0 )
d2 r µ
+ 3 r = ad
dt2
r
µ
d2 rosc
+ 3 rosc = 0
2
dt
rosc
we can write
v(t) = vosc (t) + ν(t)
3 µ
µ rosc
d2 δ
+ 3 δ = 3 1 − 3 r + ad
dt2
rosc
rosc
r
with the initial conditions
δ (t 0 ) = 0
and
dδ = ν (t0 ) = 0
dt t=t0
Coping with Numerical Accuracy for Small Deviations
When r ≈ rosc , we can define
(δ + 2rosc ) · δ
q=
2
rosc
and then write
3
3
rosc
= 1 − (1 + q)− 2
3
r
q
5 q 5 · 7 q 2 5 · 7 · 9 q 3
=3· 1−
+
−
+ ···
2
2 2
2·3 2
2·3·4 2
which is used in the classical method, or,
def
f (q) = 1 −
f (q) = q
3 + 3q + q 2
3
(1 + q) 2 + (1 + q)3
as discovered by James E. Potter.
Encke’s Method
Johann Franz Encke (1791–1865)
1. Use the Lagrangian coefficients to extrapolate along the osculating orbit:
rosc (t) = F r(t0 ) + Gv(t0 )
vosc (t) = Ft r(t0 ) + Gt v(t0 )
Note: Solving Kepler’s equation is necessary to determine the coefficients.
2. Use numerical integration to propagate the deviation vector δ :
µ
µ
d2 δ
+ 3 δ = 3 f (q)r(t) + ad
2
dt
rosc
rosc
where
r = rosc + δ
3. Use periodic rectification to maintain the efficiency of the algorithm.
16.346 Astrodynamics
Lecture 23
Navigating To Mars
Introduction Figure 7 from An Introduction to the Mathematics and
Methods of Astrodynamics. Courtesy of AIAA. Used with
permission.
Navigating to the Moon
Introduction Figure 8 from An Introduction to the Mathematics and
Methods of Astrodynamics. Courtesy of AIAA. Used with permission.
16.346 Astrodynamics
Lecture 23